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JOURNALS // Trudy Instituta Matematiki i Mekhaniki UrO RAN // Archive

Trudy Inst. Mat. i Mekh. UrO RAN, 2015 Volume 21, Number 4, Pages 212–222 (Mi timm1242)

Interpolation by functions from a Sobolev space with minimum $L_p$-norm of the Laplace operator

S. I. Novikovab

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
b Institute of Mathematics and Computer Science, Ural Federal University, Ekaterinburg

Abstract: We consider an interpolation problem with minimum value of the $L_p$-norm ($1\leq p<\infty$) of the Laplace operator of interpolants for a class of interpolated sequences that are bounded in the $l_p$-norm. The data are interpolated at nodes of the grid formed by points from $\mathbb{R}^n$ with integer coordinates. It is proved that, if $1\leq p$<$n/2$, then the $L_p$-norm of the Laplace operator of the interpolant can be arbitrarily small for any sequence that is interpolated. Two-sided estimates for the $L_2$-norm of the Laplace operator of the best interpolant are found for the case $n=2$.

Keywords: interpolation, Laplace operator, Sobolev space, embedding.

UDC: 517.5

Received: 21.01.2015



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